Spin Covariance Fluctuations in the SK Model at High Temperature

TL;DR

This paper offers a new proof for the distributional convergence of spin covariance fluctuations in the SK model at high temperature, using a path expansion approach to explain the limiting distribution.

Contribution

It introduces a novel proof method based on self-avoiding path expansions, providing a clearer understanding of the limiting distribution of covariance fluctuations.

Findings

Explicit non-Gaussian distributional limit established

Path expansion explains the form of the limiting distribution

New proof technique simplifies previous approaches

Abstract

Based on \cite{H}, it is well known that the rescaled two point correlation functions \[ \sqrt{N} \langle \sigma_i ; \sigma_j\rangle = \sqrt{N} \big( \langle \sigma_i \sigma_j\rangle -\langle \sigma_i\rangle \langle \sigma_j\rangle\big) \] in the Sherrington-Kirkpatrick spin glass model with non-zero external field admit at sufficiently high temperature an explicit non-Gaussian distributional limit as $N\to \infty$. Inspired by recent results from \cite{ABSY, BSXY, BXY}, we provide a novel proof of the distributional convergence which is based on expanding $\langle \sigma_i ; \sigma_j\rangle$ into a sum over suitable weights of self-avoiding paths from vertex $i$ to $j$. Compared to \cite{H}, our key observation is that the path representation of $\langle \sigma_i ; \sigma_j\rangle$ provides a direct explanation of the specific form of the limiting distribution of $\sqrt{N} \langle \sigma_i ; \sigma_j\rangle$.